higher-order linear dynamical systems

Higher-order linear dynamical systems

Kay Henning Brodersen

Computational Neuroeconomics Group Department of Economics, University of Zurich

Machine Learning and Pattern Recognition Group Department of Computer Science, ETH Zurich

http://people.inf.ethz.ch/bkay

2

Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

The material in these slides follows: H R Wilson (1999). Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience. Oxford University Press.

3

Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

4

Theorem 4: solving higher-order systems

5

Example: a network of three nodes

Theorem 4 tells us that the states 𝐸𝑖 will all have the form:

where −8 and −3.5 + 14.7𝑖 and −3.5 − 14.7𝑖 are the three eigenvalues of the system.

6

Example: solving vs. characterizing the system

From Chapter 3 we know that the eqilibrium point of this system

• must be a spiral point (because the eigenvalues are a complex conjugate pair)

• must be asymptotically stable (because the real part of the eigenvalues is negative)

spiral point node saddle point centre

eig([-5 -10 7; 7 -5 -10; -10 7 -5]) ans = -3.5 + 14.72243i -3.5 - 14.72243i -8

Whether or not we care to solve for the constants, we can use the eigenvalues to characterize the system:

http://demonstrations.wolfram.com/TwoDimensionalLinearSystems/

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Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

8

Theorem 5: a simple necessary condition for asymptotic stability

9

Example: testing the necessary condition

To check the condition from Theorem 5, we write down the characteristic equation:

𝐴 − 𝜆𝐼 = 0

−5 −10 77 −5 −10−10 7 −5

− 𝜆1 0 00 1 00 0 1

= 0

𝜆3 + 15 𝑎1>0

𝜆2 + 285 𝑎2>0

𝜆 + 1832𝑎3>0

= 0

Thus, a necessary condition for asymptotic stability is fulfilled.

10

Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

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Theorem 6: a simple equivalence of asymptotic stability

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Example: testing the equivalence condition

To evaluate the condition in Theorem 6, we consider three determinants:

Δ1 = 𝑎1 = 15 >0

Δ2 =𝑎1 1𝑎3 𝑎2

=15 11832 285

= 2443>0

Δ3 = 𝑎3Δ2 = 1832 ⋅ 2443 = 4475576>0

Thus, the sufficient condition for asymptotic stability is fulfilled.

time [s]

13

Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

14

Theorem 7: how to check for oscillations

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Example: checking for oscillations

To evaluate whether this system will produce oscillations, we consider two determinants:

Δ1 = 𝑎1 = 15 >0

Δ2 =𝑎1 1𝑎3 𝑎2

=15 11832 285

= 2443=0

Thus, this dynamical system will not produce oscillations. (Note that we knew this already

from the fact that the system had an asymptotically stable equilibrium point.)

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Example: enforcing oscillations (1)

To find out under what conditions this system will produce oscillations, we require that:

Δ1 = 𝑎1 > 0

Δ2 =𝑎1 1𝑎3 𝑎2

= 0

The above coefficients are obtained from the characteristic equation:

𝐴 − 𝜆𝐼 = 0

𝜆3 + 15 𝑎1

𝜆2 + 21𝑔 + 75𝑎2

𝜆 + 𝑔3 + 105𝑔 − 218𝑎3

= 0

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Example: enforcing oscillations (2)

Thus, we wish to find 𝑔 such that:

Δ1 = 15 > 0

Δ2 =15 1

𝑔3 + 105𝑔 − 218 21𝑔 + 75= 0

To satsify the second equation, we require that:

15 21𝑔 + 75 − 𝑔3 + 105𝑔 − 218 = 0

𝑔3 − 210𝑔 − 1343 = 0

This equation has 1 real solution: 𝑔 = 17.

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Example: enforcing oscillations (3)

We can use MATLAB to do the above calculations:

>> A = '[-5 -g 7; 7 -5 -g; -g 7 -5]'; >> routh_hurwitz(A) g = 17.0000 Characteristic_Eqn = 1.0e+03 * 0.0010 0.0150 0.4320 6.4800 EigenValues = 0 +20.7846i 0 -20.7846i -15.0000 RHDeterminants = 15.0000 0 0.0000 ans = Solution oscillates around equilibrium point, which is a center.

Thus, our system is described by:

𝑑

𝑑𝑡

𝐸1𝐸2𝐸3

=−5 −17 77 −5 −17−17 7 −5

𝐸1𝐸2𝐸3

-17

The states have the general form:

𝐸𝑘 = 𝐴𝑒−15𝑡 + 𝐵 cos 20.78𝑡

+𝐶 sin 20.78𝑡

Oscillation frequency: 3.3 Hz

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Example: enforcing oscillations (4)

time [s]

-17

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Outline

1 Solving higher-order systems

2 Asymptotic stability: necessary condition

3 Asymptotic stability: necessary and sufficient condition

4 Oscillations

5 Delayed feedback

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Can oscillations even occur in a simpler two-region network? We consider a simple negative feedback loop:

Oscillations in a two-region network?

𝐸 𝐼

𝑏/𝜏𝐼

−𝑎/𝜏𝐸 −𝐸/𝜏𝐸 −𝐼/𝜏𝐼

The characteristic equation is:

−1/𝜏𝐸 −𝑎/𝜏𝐸𝑏/𝜏𝐼 −1/𝜏𝐼

− 𝜆𝐼 = 0

The equation is satisfied by a complex conjugate pair of eigenvalues:

𝜆 = −1

2

1

𝜏𝐸+1

𝜏𝐼±𝜏𝐸 − 𝜏𝐼

2 − 4𝑎𝑏𝜏𝐸𝜏𝐼2𝜏𝐸𝜏𝐼

Since the real parts are negative, all solutions must be decaying exponential functions of time, and so oscillations are impossible in this model. But this does not preclude oscillations in second-order systems in general…

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Consider a two-region feedback loop with delayed inhibitory feedback:

Delayed feedback: an approximation

𝐸 𝐼

Systems with true delays become exceptionally complex. Instead, we can approximate the effect of delayed feedback by introducing an additional node:

𝐸 𝐼

Δ

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Let us specify concrete numbers for all connection strengths, except for the delay:

Delayed feedback: enforcing oscillations

𝐸 𝐼

Δ

8/50

−1/5 −1/10 −1/50

−1/𝛿

1/𝛿

Is there a value for 𝛿 that produces oscillations?

>> routh_hurwitz('[-1/10 0 -1/5; 8/50 -1/50 0; 0 1/g -1/g]',10) g = 7.6073 Solution oscillates around equilibrium point, which is a center.

time [ms]

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Summary

Preparations write down system dynamics in normal form

write down characteristic equation turn into coefficients form

What are we interested in?

Want to fully solve the system

apply theorem 5 (necessary condition)

combine with initial conditions to solve for constants 𝐴𝑛

Want to quickly check for asymptotic stability

Want to check for, or enforce, oscillations

apply theorem 4 to find eigenvalues 𝜆 (may be computationally expensive)

condition not satisfied – not stable

condition satisfied

apply theorem 6 (sufficient condition)

condition not satisfied – not stable

condition satisfied – system is stable

apply theorem 7 (necessary and sufficient condition)